Hi,
The iphone sitename is a bit of a mystery, but I think you can get what you want.
Each type of phone browser has its own template and positions can be either unique, such as mj_pda_header or shared between the templates, such as mj_all_header. You can have multiple copies of the module mod_mj_header, at the moment there is one, and it causes the problem you describe, which does either text or image for all templates.
If you go to Joomla admin back end
Extensions > Module Manager
make a copy of the Header Module
change name so it says Header PDA Module (so you can find it again)
edit it so position is mj_pda_header
module parameters show image
enable
save
Then edit the original Header Module
change name so it says Header iphone Module
edit it so position is mj_iphone_header (you may have to overwrite another existing position as it does not appear in the drop down)
module parameters show text
enable
save
repeat for mj-imode_header, mj_wap_header
I found that the iphone showed the site name without having to have a specific header module. If I had the module, then I would get the site name in two places on the iphone home page. I'm sure you can experiment for your own site to see what is best.
Hope that helps.